The author demonstrates that the Dirichlet series representation of the Riemann zeta function converges geometrically at the roots in the critical strip. The Dirichlet series parts of the Riemann zeta function diverge everywhere in the critical strip. It has therefore been assumed for at least 150 years that the Dirichlet series representation of the zeta function is useless for characterization of the non-trivial roots. The author shows that this assumption is completely wrong. Reduced, or simplified, asymptotic expansions for the terms of the zeta function series parts are equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument, constraining the real parts of the roots of both functions to the critical line. Hence, the Riemann hypothesis is correct. Formulae are derived and solved numerically, yielding highly accurate values of the imaginary parts of the roots of the zeta function.

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Samuel W. Gilbert holds a B.S. degree in chemical engineering from the University of Texas at Austin, and M.S. and Ph.D. degrees in chemical engineering in the field of molecular thermodynamics from the University of Illinois at Urbana-Champaign. He has developed mathematical models and optimization algorithms for Eastman Kodak Research Laboratories in Rochester, NY, and for Exxon Research and Engineering Company in Florham Park, NJ. He currently owns and manages an independent wealth advisory practice in Vienna, VA. Dr. Gilbert lives in northern Virginia with his wife, daughter and son.

This book is concerned with the geometric convergence of the Dirichlet series representation of the Riemann zeta function at its roots in the critical strip. The objectives are to understand why non-trivial roots occur in the Riemann zeta function, to define the roots mathematically, and to resolve the Riemann hypothesis.

The Dirichlet infinite series parts of the Riemann zeta function diverge everywhere in the critical strip. Therefore, it has always been assumed that the Dirichlet series representation of the zeta function is useless for characterization of the roots in the critical strip. In this work, it is shown that this assumption is completely wrong.

The Dirichlet series representation of the Riemann zeta function diverges algebraically everywhere in the critical strip. However, the Dirichlet series representation does, in fact, converge at the roots in the critical strip-and only at the roots in the critical strip in a special geometric sense. Although the Dirichlet series parts of the zeta function diverge both algebraically and geometrically everywhere in the critical strip, at the roots of the zeta function, the parts are geometrically equivalent and their geometric difference is identically zero.

At the roots of the Riemann zeta function, the two Dirichlet infinite series parts are coincidently divergent and are geometrically equivalent. The roots of the zeta function are the only points in the critical strip where infinite summation and infinite integration of the terms of the Dirichlet series parts are geometrically equivalent. Similarly, the roots of the zeta function with the real part of the argument reflected in the critical strip are the only points where infinite summation and infinite integration of the terms of the Dirichlet series parts with reflected argument are geometrically equivalent.

Reduced, or simplified, asymptotic expansions for the terms of the Riemann zeta function series parts at the roots, equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument at the roots, constrain the values of the real parts of both arguments to the critical line. Hence, the Riemann hypothesis is correct.

At the roots of the zeta function in the critical strip, the real part of the argument is the exponent, and the real and imaginary parts combine to constitute the coefficients of proportionality in geometrical constraints of the discrete partial sums of the series terms by a common, divergent envelope. Values of the imaginary parts of the first 50 roots of the Riemann zeta function are calculated using derived formulae with 80 correct significant figures using a laptop computer. The first five imaginary parts of the roots are:

14.134725141734693790457251983562470270784257115699243175685567460149963429809256...

21.022039638771554992628479593896902777334340524902781754629520403587598586068890...

25.010857580145688763213790992562821818659549672557996672496542006745092098441644...

30.424876125859513210311897530584091320181560023715440180962146036993329389333277...

32.935061587739189690662368964074903488812715603517039009280003440784815608630551...

It is further demonstrated that the derived formulae yield calculated values of the imaginary parts of the roots of the Riemann zeta function with more than 330 correct significant figures.

continued..

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